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Math 290 Section 2 and 4 – Midterm Exam 2
Thursday, Feb 26 through Monday, Mar 2, 2015
Professor: David Cardon, 326 TMCB, Campus Ext. 2-4863
Instructions:
• Questions 1–10 are true-false worth 3 points each. On the bubble sheet, mark A for true, B
for false.
• Questions 11–20 are multiple choice worth 3 points each. Mark the correct answer on your
bubble sheet.
• Questions 21–25 are written response questions worth 8 points each. Neatly write your
solutions directly on the exam paper. To receive full credit you must provide complete and
correct explanations.
• In written response questions words like find, show, solve, determine, or prove mean that you
should give complete explanations of the reasoning involved in the finding, showing, solving,
determining, or proving.
• Notes, books, and calculators are not allowed.
• No time limit.
For instructor use only:
TF & MC
21
22
23
24
25
Total
/60
/8
/8
/8
/8
/8
True-False Questions
1. Every subset of the natural numbers has a least element.
2. There exist a rational number x and an irrational number y such that x + y is rational.
3. Let R be a relation defined on a set A. If a R b and a R c, but b R c, then R can NOT be
transitive.
4. A relation R is defined on Z by a R b if a ≡ b (mod 2) and a ≡ b (mod 3). The relation R is
reflexive.
5. The set { n12 : n ∈ N} has a least element.
6. The union of two equivalence relations on a nonempty set A is an equivalence relation.
7. The principle of mathematical induction can be proved using the well-ordering principle for
the natural numbers.
8. There exists an integer a such that ab ≡ 0 (mod 7) for all b ∈ Z.
9. For an equivalence relation R defined on a set A and an element a ∈ A, the equivalence class
[a] is a subset of the power set of A.
10. In Z5 , [3] + [4] = [2].
Multiple Choice Section
11. Which of the following sets is well-ordered?
(a) Q.
(b) {3n − 1 : n ∈ N}.
(c) The set of nonnegative real numbers.
(d) The set of all even integers.
(e) The set of nonpositive integers.
12. In a proof by minimum counter-example for establishing the truth of the statement “∀n ∈
N, P (n)”, we would most likely start with
(a) Assume P (n) is false for all n ∈ N.
(b) Assume P (n) is false for some n ∈ N.
(c) Assume ∼ P (n) is true for all n ∈ N.
(d) Assume P (n) is true for all n ∈ N.
(e) Assume ∼ P (n) is false for all n ∈ N.
(f) None of the above.
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13. For the set A = {a, b, c} a relation R is defined by
R = {(a, b), (a, c), (b, b), (b, c), (c, b)}
Choose the most complete correct statement:
(a) R is reflexive.
(b) R is symmetric.
(c) R is transitive.
(d) R is both symmetric and transitive.
(e) R is both transitive and reflexive.
(f) None of the above.
14. An equivalence relation is defined on the set B = {1, 2, 3, 4, 5} by
R = {(1, 1), (1, 4), (1, 5), (2, 2), (2, 3), (3, 2), (3, 3), (4, 1), (4, 4), (4, 5), (5, 1), (5, 4), (5, 5)}.
Which of the following is an equivalence class?
(a) {1, 2}.
(b) {1, 2, 5}
(c) {1, 2, 3, 4, 5}
(d) {1, 2, 4, 5}
(e) {2, 3}
15. An equivalence relation R on Z defined by a R b if 5a ≡ 2b (mod 3). Which of the following
is an equivalence class of R?
(a) The set {x ∈ Z : x = 3y for some y ∈ Z}.
(b) The even integers.
(c) The odd integers.
(d) The set {x2 : x ∈ Z}.
(e) None of the above.
16. A relation R is defined on the integers by a R b if |a − b| ≤ 3. Choose the most complete
correct statement below:
(a) R is reflexive.
(b) R is symmetric.
(c) R is transitive.
(d) R is reflexive and symmetric.
(e) R is reflexive and transitive.
(f) R is symmetric and transitive.
(g) R is an equivalence relation.
(h) None of the above.
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17. Evaluate the proof of the following statement:
Result: For all n ∈ N, 3 | (4n − 1).
Proof: First note that for n = 1, 41 − 1 = 3 which is clearly divisible by 3. Next suppose that
for k ∈ N then 4k − 1 = 3m for some m ∈ N. Then
4k+1 − 1 = 4(4k ) − 1
= 4(4k − 1) − 1 + 4
= 4(3m) + 3
(by the induction hypothesis)
= 3(4m + 1),
so that 4k+1 − 1 is divisible by 3.
By the Principle of Mathematical Induction 3 | (4n − 1) for all n ∈ N.
(a) The theorem is false but the proof is correct.
(b) The proof contains arithmetic mistakes which make it incorrect.
(c) The proof incorrectly assumes what it is trying to prove.
(d) The proof is a correct proof of the stated result.
(e) None of the above.
18. Let R be the equivalence relation on Z3 defined by [a] R [b] if and only if [a]2 = [b]2 . Which
of the following is an equivalence class resulting from this relation?
(a) Z3
(b) {[0], [1]}.
(c) {[0], [2]}
(d) {[1], [2]}
(e) None of the above.
19. A proof by minimum counterexample relies on what property of the natural numbers (or the
set under consideration)?
(a) The numbers are all positive.
(b) The set has infinite cardinality.
(c) The well-ordering principle.
(d) Associativity.
(e) None of the above.
20. Which of the following is not a relation on R?
(a) A subset of R containing 2 real numbers.
(b) A set containing a single point (a, b), where both a and b are integers.
(c) The set {(r, s) ∈ R × R : r < s}.
(d) The set {(a, b) ∈ R × R : a2 + b3 = 1}.
(e) All of the above are relations on R.
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Written Answer Section
21. Prove by induction that
n
X
(3j + 2) =
j=0
for all integers n ≥ 0.
5
(n + 1)(3n + 4)
2
22. Prove that for all integers n ≥ 1, 5 | (8n − 3n ).
[Note: This problem can be correctly solved in several different ways.]
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23. Prove that for all n ∈ N, n2 6≡ 2 (mod 3).
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24. Conjucture a formula for the nth term in the sequence a1 , a2 , a3 , . . . , an , . . . defined recursively
by
a1 = 1
a2 = 2
an = an−1 + 2an−2
for n > 2.
Then use proof-by-induction to verify that your conjecture is true.
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25. (a) A relation R is defined on the real numbers R by a R b if a − b ∈ Z. Prove that R is an
equivalence relation.
√
(b) Use set-builder notation to describe the equivalence class [ 7] for the relation R in
Part (a).
√
(The final answer should express [ 7] simply and should not involve either the symbol
R or ellipses · · · .)
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